A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions
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- by R. Bruce Kellogg, Torsten Linss and Martin Stynes PDF
- Math. Comp. 77 (2008), 2085-2096 Request permission
Abstract:
An elliptic system of $M (\ge 2)$ singularly perturbed linear reaction-diffusion equations, coupled through their zero-order terms, is considered on the unit square. This system does not in general satisfy a maximum principle. It is solved numerically using a standard difference scheme on tensor-product Bakhvalov and Shishkin meshes. An error analysis for these numerical methods shows that one obtains nodal $O(N^{-2})$ convergence on the Bakhvalov mesh and $O(N^{-2}\ln ^2 N)$ convergence on the Shishkin mesh, where $N$ mesh intervals are used in each coordinate direction and the convergence is uniform in the singular perturbation parameter. The analysis is much simpler than previous analyses of similar problems, even in the case of a single reaction-diffusion equation, as it does not require the construction of an elaborate decomposition of the solution. Numerical results are presented to confirm our theoretical error estimates.References
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Additional Information
- R. Bruce Kellogg
- Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
- Email: rbmjk@windstream.net
- Torsten Linss
- Affiliation: Institut für Numerische Mathematik, Technische Univeristät, Dresden, Germany
- Email: torsten.linss@tu-dresden.de
- Martin Stynes
- Affiliation: Department of Mathematics, National University of Ireland, Cork, Ireland
- Email: m.stynes@ucc.ie
- Received by editor(s): May 24, 2007
- Published electronically: March 14, 2008
- Additional Notes: The research of the first author was supported by the Boole Centre for Research in Informatics at the National University of Ireland, Cork, and by the Science Foundation Ireland under the Basic Research Grant Programme 2004 (Grants 04/BR/M0055, 04/BR/M0055s1)
The research of the second author was supported by the Boole Centre for Research in Informatics at the National University of Ireland, Cork and by the ZIH at TU Dresden - © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 2085-2096
- MSC (2000): Primary 65N06, 65N15, 65N50; Secondary 35J45
- DOI: https://doi.org/10.1090/S0025-5718-08-02125-X
- MathSciNet review: 2429875